A mean value result involving the fourth moment of $|\zeta(1/2+it)|$

Abstract

If $(k,\ell)$ is an exponent pair such that $k+\ell<1$, then we have $$ \int_1^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^2dt \ll_\epsilon T^{1+\epsilon}\quad(\sigma > \min({5\over6},\max(\ell-k, {5k+\ell\over4k+1})), $$ while if $(k,\ell)$ is an exponent pair such that $3k+\ell<1$, then we have $$ \int_1^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^4dt \ll_\epsilon T^{1+\epsilon}\quad(\sigma > {11k+\ell+1\over8k+2}). $$